Triangle calculator SSA

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Triangle has two solutions with side c=0.92220152767 and with side c=0.39877157529

#1 Acute scalene triangle.

Sides: a = 1.06   b = 0.87   c = 0.92220152767

Area: T = 0.38224356385
Perimeter: p = 2.85220152767
Semiperimeter: s = 1.42660076384

Angle ∠ A = α = 72.46330032977° = 72°27'47″ = 1.26547179934 rad
Angle ∠ B = β = 51.5° = 51°30' = 0.89988445648 rad
Angle ∠ C = γ = 56.03769967023° = 56°2'13″ = 0.97880300954 rad

Height: ha = 0.72215766763
Height: hb = 0.87991623872
Height: hc = 0.83295646463

Median: ma = 0.72329149917
Median: mb = 0.89331019456
Median: mc = 0.85330662092

Inradius: r = 0.26881862482
Circumradius: R = 0.5565833716

Vertex coordinates: A[0.92220152767; 0] B[0; 0] C[0.66598655148; 0.83295646463]
Centroid: CG[0.52772935972; 0.27765215488]
Coordinates of the circumscribed circle: U[0.46110076384; 0.3110520655]
Coordinates of the inscribed circle: I[0.55660076384; 0.26881862482]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.5376996702° = 107°32'13″ = 1.26547179934 rad
∠ B' = β' = 128.5° = 128°30' = 0.89988445648 rad
∠ C' = γ' = 123.9633003298° = 123°57'47″ = 0.97880300954 rad




How did we calculate this triangle?

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 1.06 ; ; b = 0.87 ; ; c = 0.92 ; ;

1. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 1.06+0.87+0.92 = 2.85 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2.85 }{ 2 } = 1.43 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1.43 * (1.43-1.06)(1.43-0.87)(1.43-0.92) } ; ; T = sqrt{ 0.15 } = 0.38 ; ;

4. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.38 }{ 1.06 } = 0.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.38 }{ 0.87 } = 0.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.38 }{ 0.92 } = 0.83 ; ;

5. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 1.06**2-0.87**2-0.92**2 }{ 2 * 0.87 * 0.92 } ) = 72° 27'47" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 0.87**2-1.06**2-0.92**2 }{ 2 * 1.06 * 0.92 } ) = 51° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 0.92**2-1.06**2-0.87**2 }{ 2 * 0.87 * 1.06 } ) = 56° 2'13" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.38 }{ 1.43 } = 0.27 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1.06 }{ 2 * sin 72° 27'47" } = 0.56 ; ;





#2 Obtuse scalene triangle.

Sides: a = 1.06   b = 0.87   c = 0.39877157529

Area: T = 0.16549654639
Perimeter: p = 2.32877157529
Semiperimeter: s = 1.16438578765

Angle ∠ A = α = 107.5376996702° = 107°32'13″ = 1.87768746602 rad
Angle ∠ B = β = 51.5° = 51°30' = 0.89988445648 rad
Angle ∠ C = γ = 20.96330032977° = 20°57'47″ = 0.36658734287 rad

Height: ha = 0.31112555923
Height: hb = 0.37992309516
Height: hc = 0.83295646463

Median: ma = 0.42202843205
Median: mb = 0.67220594543
Median: mc = 0.949905508

Inradius: r = 0.14217402135
Circumradius: R = 0.5565833716

Vertex coordinates: A[0.39877157529; 0] B[0; 0] C[0.66598655148; 0.83295646463]
Centroid: CG[0.35325270893; 0.27765215488]
Coordinates of the circumscribed circle: U[0.19988578765; 0.51990439912]
Coordinates of the inscribed circle: I[0.29438578765; 0.14217402135]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 72.46330032977° = 72°27'47″ = 1.87768746602 rad
∠ B' = β' = 128.5° = 128°30' = 0.89988445648 rad
∠ C' = γ' = 159.0376996702° = 159°2'13″ = 0.36658734287 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 1.06 ; ; b = 0.87 ; ; beta = 51° 30' ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 0.87**2 = 1.06**2 + c**2 -2 * 0.87 * c * cos (51° 30') ; ; ; ; c**2 -1.32c +0.367 =0 ; ; p=1; q=-1.31973102967; r=0.3667 ; ; D = q**2 - 4pr = 1.32**2 - 4 * 1 * 0.367 = 0.274889990678 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 1.32 ± sqrt{ 0.27 } }{ 2 } ; ; c_{1,2} = 0.659865514836 ± 0.26214976191 ; ;
c_{1} = 0.922015276746 ; ; c_{2} = 0.397715752926 ; ; ; ; (c -0.922015276746) (c -0.397715752926) = 0 ; ; ; ; c>0 ; ;


Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 1.06 ; ; b = 0.87 ; ; c = 0.4 ; ;

2. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 1.06+0.87+0.4 = 2.33 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2.33 }{ 2 } = 1.16 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1.16 * (1.16-1.06)(1.16-0.87)(1.16-0.4) } ; ; T = sqrt{ 0.03 } = 0.16 ; ;

5. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.16 }{ 1.06 } = 0.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.16 }{ 0.87 } = 0.38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.16 }{ 0.4 } = 0.83 ; ;

6. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 1.06**2-0.87**2-0.4**2 }{ 2 * 0.87 * 0.4 } ) = 107° 32'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 0.87**2-1.06**2-0.4**2 }{ 2 * 1.06 * 0.4 } ) = 51° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 0.4**2-1.06**2-0.87**2 }{ 2 * 0.87 * 1.06 } ) = 20° 57'47" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.16 }{ 1.16 } = 0.14 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1.06 }{ 2 * sin 107° 32'13" } = 0.56 ; ;




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